Gaussian Beams

The complex amplitude of Gaussian beam has the form,

$$ \begin{equation} U(r) = A_{0} \dfrac{W_{0}}{W(z)} exp\left[-\dfrac{\rho^2}{W^2(z)}\right] exp\left[-jkz - jk\dfrac{\rho^2}{2R(z)} + j\zeta(z)\right] \end{equation} $$

$ \textbf{Beam width} \; \begin{equation} W(z) = \sqrt{\dfrac{\lambda z_{0}}{\pi}} \sqrt{1 + \left(\dfrac{z}{z_{0}}\right)^2} = W_{0} \sqrt{1 + \left(\dfrac{z}{z_{0}}\right)^2} \\ \end{equation} $ $ \begin{equation} \textbf{Wavefront radius of curvature} \; R(z) = z \left[1 + \left(\dfrac{z_{0}}{z}\right)^2\right] \\ \end{equation} $ $ \begin{equation} \textbf{Gouy phase} \; \zeta(z) = tan^{-1}\left(\dfrac{z}{z_{0}}\right) \\ \end{equation} $ $ \begin{equation} \textbf{Rayleigh range} \; z_{0} = \dfrac{\pi W_{0}^2}{\lambda} \\ \end{equation} $ $ \begin{equation} \textbf{Beam waist} \; W_{0} = \sqrt{\dfrac{\lambda z_{0}}{\pi}} \end{equation} $

Beam width

Beam width at y-z plane

Radius of curvature

Intensity

The intensity of the Gaussian beam $I(\rho, z) = \left| U(r) \right|^2$ is,

$$ \begin{align} I(\rho, z) &= \left| A_{0} \right|^2 \left[\dfrac{W_{0}}{W(z)}\right]^2 exp\left[-\dfrac{2\rho^2}{W^2(z)}\right] \\ &= I_{0}\left[\dfrac{W_{0}}{W(z)}\right]^2 exp\left[-\dfrac{2\rho^2}{W^2(z)}\right] \end{align} $$

Intensity along beam axis $(\rho=0)$

Intensity at the transverse plane

Assume z at 0, $z_{0}$, $2z_{0}$

Intensity at y-z plane

Hermite-Gaussian beam

The complex amplitude of Hermite-Gaussian beam has the form,

$$ \begin{equation} U_{l,m}(x,y,z) = A_{l,m} \left[ \dfrac{W_{0}}{W(z)}\right] \mathbb{G}_{l}\left[\dfrac{\sqrt{2}x}{W(z)}\right] \mathbb{G}_{m}\left[\dfrac{\sqrt{2}y}{W(z)}\right] exp\left[-jkz - jk\dfrac{x^2+y^2}{2R(z)} + j(l+m+1)\zeta(z)\right] \end{equation} $$

$ \textbf{Hermite-Gaussian function} \; \begin{equation} \mathbb{G}_{l}(u) = \mathbb{H}_{l}(u) \; exp\left(-\dfrac{u^2}{2}\right), \; l=0,1,2,... \\ \end{equation} $ $ \begin{equation} \textbf{Hermite polynomial} \; \mathbb{H}_{l+1}(u) = 2u\mathbb{H}_{l}(u) - 2l\mathbb{H}_{l-1}(u) ,\:\text{where} \; \mathbb{H}_{0}(u)=1, \; and \; \mathbb{H}_{1}(u)=2u \end{equation} $

Intensity

The intensity of the Hermite-Gaussian beam $I_{l,m} = \left| U_{l,m} \right|^2$ is,

$$ \begin{equation} I_{l,m}(x,y,z) = \left|A_{l,m}\right|^2 \left[ \dfrac{W_{0}}{W(z)}\right]^2 \mathbb{G}_{l}^2\left[\dfrac{\sqrt{2}x}{W(z)}\right] \mathbb{G}_{m}^2\left[\dfrac{\sqrt{2}y}{W(z)}\right] \end{equation} $$

Laguerre-Gaussian beams

The complex amplitude of Laguerre-Gaussian beam has the form,

$$ \begin{equation} U_{l,m}(\rho,\phi,z) = A_{l,m} \left[ \dfrac{W_{0}}{W(z)}\right] \left( \dfrac{\rho}{W(z)}\right)^{l} \mathbb{L}^{l}_{m}\left(\dfrac{2\rho^{2}}{W^{2}(z)}\right) exp\left(-\dfrac{\rho^{2}}{W^{2}(z)}\right) exp\left[-jkz - jk\dfrac{\rho^{2}}{2R(z)} \mp{jl\phi}+ j(l+2m+1)\zeta(z)\right] \end{equation} $$

$ \textbf{Generalized Laguerre polynomial} \; \begin{equation} \mathbb{L}^{l}_{m+1}(u) = \dfrac{(2m+1+l-u)\mathbb{L}^{l}_{m}(u) - (m+l)\mathbb{L}^{l}_{m-1}(u)}{m+1} , \\ \:\text{where} \; \mathbb{L}^{l}_{0}(u) = 1 \; and \; \mathbb{L}^{l}_{1}(u) = 1 + l - u \end{equation} $

Intensity

The intensity of the Laguerre-Gaussian beam $I_{l,m} = \left| U_{l,m} \right|^2$ is,

$$ \begin{equation} I_{l,m}(\rho,\phi,z) = \left|A_{l,m}\right|^{2} \left[ \dfrac{W_{0}}{W(z)}\right]^{2} \left( \dfrac{\rho}{W(z)}\right)^{2l} {\mathbb{L}^{l}_{m}}^{2}\left(\dfrac{2\rho^{2}}{W^{2}(z)}\right) exp\left(-\dfrac{2\rho^{2}}{W^{2}(z)}\right) \end{equation} $$

Bessel beams (nondiffracting beams)

The complex envelope of Bessel beam has the form,

$$ \begin{equation} A(x,y) = A_{m} J_{m}(k_{T}\rho) e^{jm\phi}, \; \text{where} \; k_{T}^2 + \beta^2 = k^2 \end{equation} $$

$ \textbf{Bessel function of the first kind and $m^{th}$ order} \; \begin{equation} J_{m}(x) = \sum_{n=0}^{\infty}{\dfrac{(-1)^{n}}{n! \Gamma(n+m+1)} \left( \dfrac{x}{2}\right)^{2n+m}} \end{equation} $

Thus, the complex amplitude is,

$$ \begin{equation} U(r) = A_{m} J_{m}(k_{T}\rho) e^{jm\phi}, \; \text{where} \; k_{T}^2 + \beta^2 = k^2 \end{equation} $$

If $m=0$, the wave has complex amplitude, $ \begin{equation} U(r) = A_{0} J_{0}(k_{T}\rho) e^{-j \beta z} \end{equation} $ ,and the intensity distribution $I(\rho, \phi, z) = \left|A_{0}\right|^{2} J_{0}^{2}(k_{T}\rho)$